The process matrix formalism[1]}, [2]}, [3]}, [4]}, [5]} is simply the quantum version of this formalism. We define a `process' to be composed of a set of agents in separate laboratories who are free to perform any local operation which is permitted by quantum mechanics, together with a `process matrix' which describes the quantum dynamics between the labs, without making any assumptions about the causal ordering of the labs. The process matrix operates as a higher-order map from a set of quantum operations to another operation - i.e. it maps the quantum operations performed by the agents in the laboratories to an operation representing the net effect of performing all of those operations in the context of the particular process. The set of valid process matrices is defined in such a way as to include all possible dynamics which will never give rise to contradictions no matter which operations the individual agents choose to perform. Again we find that there are possible dynamics which don't have a well-defined causal order but which allow agents to perform all operations allowed by quantum physics; but in the quantum case we have more possibilities for non-causal structures - we could still have CTCs, but we could also have superpositions of different causal structures, which are often characterised as `indefinite causal order.'
